Question

Find the derivative of each function. Verify some of your results by calculator. As usual, the letters \(a, b, c, \ldots\) represent constants. Derivative of a Sum. $$y=3 x^{3}+7 x^{2}-2 x+5$$

Short answer

The derivative of the function \(y = 3x^3 + 7x^2 - 2x + 5\) is \(y' = 9x^2 + 14x - 2\).

Step-by-step solution

Identify the Function

Given the function is a polynomial of the form: \(y=3x^3+7x^2-2x+5\). We need to find the derivative of this function.

Apply the Power Rule

Use the power rule for differentiation, which states that the derivative of \(x^n\) with respect to \(x\) is \(nx^{n - 1}\). Apply this rule to each term of the polynomial function.

Differentiate Each Term

Differentiate each term of the polynomial separately. For \(3x^3\), the derivative is \(9x^2\). For \(7x^2\), the derivative is \(14x\). For \(-2x\), the derivative is \(-2\). Constants like \(5\) have a derivative of \(0\).

Combine the Derivatives

Combine the derivatives of each term to form the derivative of the entire function, \(y'\) or \(\frac{dy}{dx}\). \(y' = 9x^2 + 14x - 2\).

Verify with a Calculator (Optional)

To verify the result, enter the original function into a calculator that has differentiation capabilities, and compare the calculator's result with the derived result.

Key concepts

Power Rule

When delving into the basics of calculus, one of the most fundamental techniques you'll learn is the power rule for differentiation. It simplifies the process of finding the derivatives of polynomials and is a real time-saver. For any term in the polynomial that looks like a x^n (where n is a real number), the power rule tells us that the derivative of this term will be an x^{n-1}. On a practical level, you just multiply the coefficient (a) by the exponent (n) and then decrease the exponent by one. For example, if you have 3x^3, using the power rule will give you 3*3x^{3-1}, which simplifies to 9x^2. It’s a method that streamlines differentiation and is really easy to apply!

Polynomial Differentiation

Now, let's expand our understanding to polynomial differentiation as a whole. A polynomial function is one that involves several terms with variables raised to whole number exponents, like 3x^3 + 7x^2 - 2x + 5. Differentiating a polynomial isn't much harder than applying the power rule to each term individually. You essentially break it down into smaller, more manageable pieces. Here’s a quick rundown:

• Take each term one at a time
• Apply the power rule
• Sum up all the differentiated terms

One critical thing to remember is that the derivative of a constant is zero. Why? Because constants don’t change—they have no 'rate of change,' which is what derivatives measure. So anytime you see a constant, like + 5, you can just set it aside because its derivative doesn't contribute to the overall result.

Calculus

Calculus, the mathematical study of continuous change, is a versatile and powerful tool that has applications in science, engineering, economics, and beyond. It’s divided into two main branches—differential and integral calculus. Differential calculus, which is what we engage with when finding derivatives, focuses on the rate of change of quantities. It answers questions like 'How fast is a car accelerating?' or 'At what rate is the area of a circle increasing when its radius grows?' Integral calculus, on the other hand, answers questions about the accumulation of quantities, such as 'How much total distance has the car covered?' or 'What is the total area inside the circle?' Understanding calculus early on allows students to grasp a wide range of phenomena in the natural and social worlds.

Derivative Verification

Derivative verification is an essential step to ensure that the work you've done to find a derivative is correct. To verify your solution, one approach is to use a graphical calculator or computer algebra system that can evaluate derivatives at the press of a button. After calculating the derivative by hand, inputting the same function into the calculator should yield the same result, confirming the accuracy of your work. Another method of verification is checking your answer against trusted solutions, such as those provided by your textbook or instructor. While technology can help, becoming proficient with derivative verification yourself reinforces your understanding and ensures you're not solely reliant on electronic tools. After all, truly understanding the concepts is the goal, not just arriving at the correct answer.